Remarks on an overdetermined boundary value problem. (English) Zbl 1166.35353
Summary: We modify and extend proofs of Serrin’s symmetry result for overdetermined boundary value problems from the Laplace-operator to a general quasilinear operator and remove a strong ellipticity assumption in [Philippin, in: Maximum principles and eigenvalue problems in partial differential equations, Proc. Conf., Knoxville/Tenn. 1987, Pitman Res. Notes Math. Ser. 175, 34–48 (1988; Zbl 0658.35012)] and a growth assumption in [N. Garofalo and J. L. Lewis, Am. J. Math. 111, No. 1, 9–33 (1989; Zbl 0681.35016)] on the diffusion coefficient \(A\), as well as a starshapedness assumption on \(\Omega\) in [I. Fragalà et al., Math. Z. 254, No. 1, 117–132 (2006; Zbl 1220.35077)].
MSC:
| 35N10 | Overdetermined systems of PDEs with variable coefficients |
| 35J65 | Nonlinear boundary value problems for linear elliptic equations |
| 35B35 | Stability in context of PDEs |
Keywords:
overdetermined elliptic boundary value problems; Serrin’s symmetry; general quasilinear operator; strong ellipticity assumption; starshapedness assumptionReferences:
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